Time-dependent earthquake probabilities

[1] We have attempted to provide a careful examination of a class of approaches for estimating the conditional probability of failure of a single large earthquake, particularly approaches that account for static stress perturbations to tectonic loading as in the approaches of Stein et al. ( 1997) and Hardebeck ( 2004). We have developed a general framework based on a simple, generalized rate change formulation and applied it to these two approaches to show how they relate to one another. We also have attempted to show the connection between models of seismicity rate changes applied to ( 1) populations of independent faults as in background and aftershock seismicity and ( 2) changes in estimates of the conditional probability of failure of a single fault. In the first application, the notion of failure rate corresponds to successive failures of different members of a population of faults. The latter application requires specification of some probability distribution ( density function or PDF) that describes some population of potential recurrence times. This PDF may reflect our imperfect knowledge of when past earthquakes have occurred on a fault ( epistemic uncertainty), the true natural variability in failure times, or some combination of both. We suggest two end-member conceptual single-fault models that may explain natural variability in recurrence times and suggest how they might be distinguished observationally. When viewed deterministically, these single-fault patch models differ significantly in their physical attributes, and when faults are immature, they differ in their responses to stress perturbations. Estimates of conditional failure probabilities effectively integrate over a range of possible deterministic fault models, usually with ranges that correspond to mature faults. Thus conditional failure probability estimates usually should not differ significantly for these models.